Question

The equilibrium equations in terms of total stresses formed by summing all forces on x-direction is ________

(a) \(\frac{∂σ_x}{∂x} + \frac{∂τ_{yx}}{∂y} + \frac{∂τ_{zx}}{∂z} +X=0\)

(b) \(\frac{∂τ_{xy}}{∂x} + \frac{∂σ_y}{∂y} +\frac{∂τ_{zy}}{∂z}+Y=0\)

(c) \(\frac{∂τ_{xz}}{∂x} +\frac{∂τ_{yz}}{∂y} +\frac{∂σ_z}{∂z} +Z=0\)

(d) \(\frac{∂σ_x}{∂x}+\frac{∂τ_{yx}}{∂y} +\frac{∂τ_{zx}}{∂z} = 0\)

The question was posed to me in final exam.

My question is from Elasticity in section Elements of Elasticity of Geotechnical Engineering I

Answer 1

The correct choice is (d) \(\frac{∂σ_x}{∂x}+\frac{∂τ_{yx}}{∂y} +\frac{∂τ_{zx}}{∂z} = 0\)

Easiest explanation: The equilibrium equations in terms of total stresses formed by summing all forces on x-direction will include seepage force; the body forces will be equal to those due to gravity in respective directions.

∴X=0,

The equilibrium equation obtained by summing all forces on x-direction is,

\(\frac{∂σ_x}{∂x} + \frac{∂τ_{yx}}{∂y} + \frac{∂τ_{zx}}{∂z} +X=0\)

 Therefore, the equilibrium equations in terms of total stresses formed by summing all forces on x-direction is,

\(\frac{∂σ_x}{∂x}+\frac{∂τ_{yx}}{∂y} +\frac{∂τ_{zx}}{∂z} = 0.\)